From 5bf78e09e1847646e841b9a7aa84995894ceb5b5 Mon Sep 17 00:00:00 2001 From: Andreas Tsouchlos Date: Wed, 17 Dec 2025 01:03:47 +0100 Subject: [PATCH] Add theory for part 2 --- src/2025-12-19/presentation.tex | 206 +++++++++++++++++++++++++++----- 1 file changed, 175 insertions(+), 31 deletions(-) diff --git a/src/2025-12-19/presentation.tex b/src/2025-12-19/presentation.tex index 4c53b8f..b7b32df 100644 --- a/src/2025-12-19/presentation.tex +++ b/src/2025-12-19/presentation.tex @@ -386,50 +386,194 @@ %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \subsection{Theorie Wiederholung} -% TODO: Write -% TODO: Plot -% TODO: Mention it appears regularly because of the CLT \begin{frame} \frametitle{Die Normalverteilung} - \begin{itemize} - \item TODO - \begin{align*} - f_X(x) = \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(\frac{(x - \mu)^2}{2 \sigma^2} \right) \\ - \end{align*} - \end{itemize} + \begin{columns} + \column{\kitthreecolumns} + \centering + \begin{gather*} + X \sim \mathcal{N}\mleft( \mu, \sigma^2 \mright) + \end{gather*}% + \vspace{2mm} + \begin{align*} + f_X(x) &= \frac{1}{\sqrt{2\pi \sigma^2}} \exp\left(\frac{(x - + \mu)^2}{2 \sigma^2} \right) \\[2mm] + F_X(x) &= + \vcenter{\hbox{\scalebox{1.5}[2.6]{\vspace*{3mm}$\displaystyle\int$}}}_{\hspace{-0.5em}-\infty}^{\,x} + \frac{1}{\sqrt{2\pi + \sigma^2}} \exp\left(\frac{(u - \mu)^2}{2 \sigma^2} \right) du + \end{align*} + \column{\kitthreecolumns} + \centering + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + domain=-4:4, + xmin=-4,xmax=4, + width=15cm, + height=5cm, + samples=200, + xlabel={$x$}, + ylabel={$f_X(x)$}, + xtick={0}, + xticklabels={\textcolor{KITblue}{$\mu$}}, + ytick={0}, + ] + \addplot+[mark=none, line width=1pt] + {(1 / sqrt(2*pi)) * exp(-x*x)}; + + \addplot+ [KITblue, mark=none, line width=1pt] + coordinates {(-0.5, 0.15) (0.5, 0.15)}; + \addplot+ [KITblue, mark=none, line width=1pt] + coordinates {(-0.5, 0.12) (-0.5, 0.18)}; + \addplot+ [KITblue, mark=none, line width=1pt] + coordinates {(0.5, 0.12) (0.5, 0.18)}; + \node[KITblue] at (axis cs: 0, 0.2) {$\sigma$}; + + % \addplot +[scol2, mark=none, line width=1pt] + % coordinates {(4.8, -1) (4.8, 2)}; + % \addplot +[scol2, mark=none, line width=1pt] + % coordinates {(5.2, -1) (5.2, 2)}; + % \node at (axis cs: 4.8, 3) {$S(1-\delta)$}; + % \node at (axis cs: 5.2, 3) {$S(1+\delta)$}; + \end{axis} + \end{tikzpicture} + \end{figure} + \begin{figure}[H] + \centering + \begin{tikzpicture} + \begin{axis}[ + domain=-4:4, + xmin=-4,xmax=4, + width=15cm, + height=5cm, + samples=200, + xlabel={$x$}, + ylabel={$F_X(x)$}, + xtick=\empty, + ytick={0, 1}, + ] + \addplot+[mark=none, line width=1pt] + {1 / (1 + exp(-(1.526*x*(1 + 0.1034*x))))}; + \end{axis} + \end{tikzpicture} + \end{figure} + \end{columns} \end{frame} -% TODO: Write -% TODO: Define Phi -% TODO: Phi Rechenregeln -% TODO: Define Explain use of tables +% TODO: Are Z/z notation used in the lecture? \begin{frame} \frametitle{Rechnen mithilfe der Standardnormalverteilung} + \vspace*{-15mm} + \begin{itemize} - \item Standardisierte ZV: - \begin{align*} - E(X) &= 0 \\ - V(X) &= 1 - \end{align*} - \item Standardisierung einer ZV: - \begin{align*} - \widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}} - \end{align*} - \item TODO: + \item Die Standardnormalverteilung + \end{itemize} + + \begin{minipage}{0.48\textwidth} + \centering + \begin{gather*} + Z \sim \mathcal{N} (0,1) \\[4mm] + \Phi(z) := F_Z(z) = P(Z \le z) \\ + \Phi(-z) = 1 - \Phi(z) + \end{gather*} + \end{minipage}% + \begin{minipage}{0.48\textwidth} + \centering + \begin{tabular}{|c|c||c|c||c|c|} + \hline + $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ \\ + \hline + \hline + 0{,}00 & 0{,}500000 & 0{,}10 & 0{,}539828 & 0{,}20 & 0{,}579260 \\ + 0{,}02 & 0{,}507978 & 0{,}12 & 0{,}547758 & 0{,}22 & 0{,}587064 \\ + 0{,}04 & 0{,}515953 & 0{,}14 & 0{,}555670 & 0{,}24 & 0{,}594835 \\ + 0{,}06 & 0{,}523922 & 0{,}16 & 0{,}563559 & 0{,}26 & 0{,}602568 \\ + 0{,}08 & 0{,}531881 & 0{,}18 & 0{,}571424 & 0{,}28 & 0{,}610261 \\ + \hline + \end{tabular}\\ + \end{minipage} + + \pause + \begin{itemize} + \item Standardisierte ZV \begin{gather*} - X \sim \mathcal{N}(\mu, \sigma^2) \\[5mm] - P(X \le a) = P\bigg(\underbrace{\frac{X - \mu}{\sigma}}_{:= Z \sim \mathcal{N}(0,1)} \le \frac{a - \mu}{\sigma}\bigg) - = P\bigg(Z \le \frac{a - \mu}{\sigma}\bigg) = \Phi\mleft( \frac{a - \mu}{\sigma} \mright) + \begin{array}{cc} + E(X) &= 0 \\ + V(X) &= 1 + \end{array} + \hspace{45mm} + \text{Standardisierung: } \hspace{5mm} + \widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}} + = \frac{X - \mu}{\sigma} \end{gather*} \end{itemize} + + \vspace*{5mm} + + \pause + \begin{lightgrayhighlightbox} + Rechenbeispiel + \begin{gather*} + X \sim \mathcal{N}(\mu = 1, \sigma^2 = 0{,}5^2) \\[2mm] + P\left(X \le 1{,}12 \right) + = P\left(\frac{X - 1}{0{,}5} \le \frac{1{,}12 - 1}{0{,}5}\right) + = P\left(\frac{X - 1}{0{,}5} \le + 0{,}24\right) = \Phi\left(0{,}24\right) = 0{,}594835 + \end{gather*} + \end{lightgrayhighlightbox} \end{frame} -% TODO: Write -% TODO: Include Phi table? +% TODO: Are Z/z notation used in the lecture? \begin{frame} \frametitle{Zusammenfassung} + + \vspace*{-15mm} + + \begin{columns}[t] + \column{\kitthreecolumns} + \centering + \begin{greenblock}{Standardnormalverteilung} + \vspace*{-10mm} + \begin{gather*} + Z \sim \mathcal{N} (0,1) \\[4mm] + \Phi(z) := F_Z(z) = P(Z \le z) \\ + \Phi(-z) = 1 - \Phi(z) + \end{gather*} + \end{greenblock} + \column{\kitthreecolumns} + \centering + \begin{greenblock}{Standardisierung} + \vspace*{-10mm} + \begin{gather*} + \widetilde{X} = \frac{X - E(X)}{\sqrt{V(X)}} + = \frac{X - \mu}{\sigma} + \end{gather*} + \end{greenblock} + \end{columns} + + \vspace{5mm} + + \begin{table} + \centering + % \cdots + \begin{tabular}{|c|c||c|c||c|c||c|c|} + \hline + $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ & $z$ & $\Phi(z)$ \\ + \hline + \hline + 1{,}40 & 0{,}919243 & 2{,}80 & 0{,}997445 & 3{,}00 & 0{,}998650 & 4{,}20 & 0{,}999987 \\ + 1{,}42 & 0{,}922196 & 2{,}82 & 0{,}997599 & 3{,}02 & 0{,}998736 & 4{,}22 & 0{,}999988 \\ + 1{,}44 & 0{,}925066 & 2{,}84 & 0{,}997744 & 3{,}04 & 0{,}998817 & 4{,}24 & 0{,}999989 \\ + 1{,}46 & 0{,}927855 & 2{,}86 & 0{,}997882 & 3{,}06 & 0{,}998893 & 4{,}26 & 0{,}999990 \\ + 1{,}48 & 0{,}930563 & 2{,}88 & 0{,}998012 & 3{,}08 & 0{,}998965 & 4{,}28 & 0{,}999991 \\ + \hline + \end{tabular} + % \cdots + \end{table} \end{frame} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% @@ -481,7 +625,7 @@ \column{\kitthreecolumns} \centering \pause \begin{gather*} - X \sim \mathcal{N} \mleft( \mu = 0{,}5, \sigma = 0{,}07^2 \mright) + X \sim \mathcal{N} \mleft( \mu = 0{,}5, \sigma^2 = 0{,}07^2 \mright) \end{gather*} \begin{align*} P(E_\text{a}) &= P \Big( \big( X < S(1-\delta) \big) @@ -511,8 +655,8 @@ \addplot+[mark=none, line width=1pt] {1 / sqrt(2*3.1415*0.07*0.07) * exp(-(x - 5)*(x-5)/(2*0.07*0.07))}; - \addplot +[scol2, mark=none, line width=1pt] coordinates {(4.8, -1) (4.8, 2)}; - \addplot +[scol2, mark=none, line width=1pt] coordinates {(5.2, -1) (5.2, 2)}; + \addplot +[KITblue, mark=none, line width=1pt] coordinates {(4.8, -1) (4.8, 2)}; + \addplot +[KITblue, mark=none, line width=1pt] coordinates {(5.2, -1) (5.2, 2)}; \node at (axis cs: 4.8, 3) {$S(1-\delta)$}; \node at (axis cs: 5.2, 3) {$S(1+\delta)$};